A226178 -id:A226178 - cp's OEIS frontend

A340707 a(n) = (prevprime(2^n) + nextprime(2^n))/2 - 2^n where prevprime(n) = A151799(n) and nextprime(n) = A151800(n).

0, 1, -1, 2, 0, 1, -2, 3, 2, -2, 0, 8, 12, -8, -7, 14, -1, 10, 2, 4, 6, -3, 20, -2, 5, -5, -27, 4, -16, 5, 5, 4, -8, 11, 13, -8, -19, 8, -36, 3, 2, -14, -5, 2, -3, -55, -19, -6, 14, -54, -13, -53, 63, -26, 38, -2, 21, 38, -30, 7, 39, 2, -23, 41, 2, -8, 5, 5, -5, -110

Offset: 2

Hugo Pfoertner, Jan 29 2021

sign

a(n) > 0 if the difference nextprime(2^n) - 2^n = A013597(n) is greater than the difference 2^n - previousprime(2^n) = A013603(n).

			a(4) = -1: 2^4 = 16, (13 + 17 - 32)/2 = -1;
a(5) = 2: 2^5 = 32, (31 + 37 - 64)/2 = 2;
a(6) = 0: 2^6 = 64, (61 + 67 - 128)/2 = 0.

Hugo Pfoertner, Table of n, a(n) for n = 2..5000 (from T. D. Noe's b-files in A013597 and A013603).

Cf. A151799, A151800, A013597, A013603, A058249, A226178.

a:= (p-> (nextprime(p)+prevprime(p))/2-p)(2^n):
seq(a(n), n=2..75);  # Alois P. Heinz, Jan 29 2021

Array[(NextPrime[2^#] + NextPrime[2^#, -1] - 2^(# + 1))/2 &, 60, 2] (* Michael De Vlieger, Aug 07 2022 *)

for(k=2,71,my(p2=2^k,pp=precprime(p2),pn=nextprime(p2));if(print1((pp+pn-2*p2)/2", ")))

a(n) = (A013597(n) - A013603(n))/2.

a(A226178(n)) = 0.

Name made more precise by Peter Luschny, Aug 08 2022

A356434 Prime nearest to 2^n. In case of a tie, choose the larger.

Original entry on oeis.org

2, 2, 5, 7, 17, 31, 67, 127, 257, 509, 1021, 2053, 4099, 8191, 16381, 32771, 65537, 131071, 262147, 524287, 1048573, 2097143, 4194301, 8388617, 16777213, 33554467, 67108859, 134217757, 268435459, 536870909, 1073741827, 2147483647, 4294967291, 8589934583

Offset: 0

Peter Munn, Aug 07 2022

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000

A117387 differs by preferring the smaller prime in the case of a tie, which occurs when n is in A226178.

Cf. A014210, A014234, A340707.

Join[{2,2},Table[Max[Nearest[{NextPrime[2^n,-1],NextPrime[2^n]},2^n]],{n,2,40}]] (* Harvey P. Dale, Feb 19 2023 *)

from sympy import prevprime, nextprime
def A356434(n): return (r if (m:=nextprime(k:=1< (k<<1)-(r:=prevprime(k)) else m) if n>1 else 2 # Chai Wah Wu, Aug 08 2022

a(0) = 2; for n >= 1, if A014210(n) + A014234(n) > 2^(n+1) then a(n) = A014234(n), otherwise a(n) = A014210(n).

A372263 Least odd prime factor of the n-th sum of two consecutive primes, A001043(n) = prime(n) + prime(n+1), or 2 if there is no odd prime factor.

Original entry on oeis.org

5, 2, 3, 3, 3, 3, 3, 3, 13, 3, 17, 3, 3, 3, 5, 7, 3, 2, 3, 3, 19, 3, 43, 3, 3, 3, 3, 3, 3, 3, 3, 67, 3, 3, 3, 7, 5, 3, 5, 11, 3, 3, 3, 3, 3, 5, 7, 3, 3, 3, 59, 3, 3, 127, 5, 7, 3, 137, 3, 3, 3, 3, 3, 3, 3, 3, 167, 3, 3, 3, 89, 3, 5, 47, 3, 193, 3, 3, 3, 3, 3, 3, 3, 109, 3, 223

Offset: 1

M. F. Hasler, Apr 24 2024

nonn

Since the sum of any two primes > 2 is even, we rather consider odd prime factors.
Can it be proved or disproved that there are primes that occur only finitely many times (or never) in this sequence? If so, which is the smallest such prime?
From Robert Israel, Dec 29 2024: (Start)
Dickson's conjecture implies that every odd prime occurs infinitely many times in the sequence.
a(n) = 2 if and only if n = A000720(2^k) where k is in A226178. (End)

			Sums of two consecutive primes are given as s(n) = A001043(n). The least odd prime factor (or 2 if there's no odd prime factor) of these terms is a(n):
n = 1, 2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14,  15,  16,  17,  18, ...
s = 5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, 78, 84, 90, 100, 112, 120, 128, ...
a = 5, 2,  3,  3,  3,  3,  3,  3, 13,  3, 17,  3,  3,  3,   5,   7,   3,   2, ...
Also, a(21) = spf(152) = 19; a(23) = spf(172) = 43; a(32) = spf(268) = 67, ...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001043 (sums of two consecutive primes),  A078701 (least odd prime divisor), A020639 (spf: least prime factor), A000265 (odd part of n), A000079 (powers of 2).
Cf. A024677 (spf of A024675(n) = A001043(n)/2).
Cf. A226178.

f:= proc(n) subs(infinity=2,min(numtheory:-factorset(ithprime(n)+ithprime(n+1)) minus {2})) end proc:
map(f, [$1..100]); # Robert Israel, Dec 29 2024

apply( {a(n) = max(A078701(A001043(n)), 2)}, [1..99])
/* a "self-contained" but less efficient definition:
a(n) = factor(max((n=prime(n)+prime(n+1))>>valuation(n,2),2))[1,1] */

a(n) = max(A078701(A001043(n)), 2) = A020639(max(A000265(A001043(n)), 2)), where A000265(m) > 2 unless m is in A000079.

cp's OEIS Frontend

A340707 a(n) = (prevprime(2^n) + nextprime(2^n))/2 - 2^n where prevprime(n) = A151799(n) and nextprime(n) = A151800(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A356434 Prime nearest to 2^n. In case of a tie, choose the larger.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Formula

A372263 Least odd prime factor of the n-th sum of two consecutive primes, A001043(n) = prime(n) + prime(n+1), or 2 if there is no odd prime factor.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula